Some simple methods for the unique assignment of a density distribution to a harmonic function
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Date
1974-08
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
One to one relationships are established between functions harmonic outside a sphere and density functions (1) with support equal to the sphere and (2) having the property that the density functions 𝛒, multiplied by a positive function of the distance from the center of the sphere, are harmonic, i.e., (*) ∆ (f(r) • 𝛒) = 0, f(r) > 0 for 0 ≤ r ≤ R. The relationship is established by specifying the relation for each external solid spherical harmonic Venn of degree n and order m. The Poisson equation is first used to obtain a density function, equal to a distribution with support in the center of the sphere. This density is then spread out inside the whole sphere. As spreading operators arc used the identity operators on Hilbert spaces of density functions fulfilling (*). The derived relations may be used to assign a density distribution to the harmonic part of the potential of the Earth, and a covariance function of a density anomaly distribution to the covariance function of the anomalous potential of the Earth. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]